logarithmic derivative

Given a function $f$, the quantity $f^{\prime }/f$ is known as the logarithmic derivative of $f$. This name comes from the observation that, on account of the chain rule,

$${(\log f)}^{\prime }=f^{\prime }{\log }^{\prime }(f)=f^{\prime }/f.$$

The logarithmic derivative has several basic properties which make it useful in various contexts.

The logarithmic derivative of the product of functions is the sum of their logarithmic derivatives. This follows from the product rule:

$$\frac{{(fg)}^{\prime }}{fg}=\frac{f{g}^{\prime }+f^{\prime }g}{fg}=\frac{f^{\prime }}{f}+\frac{g^{\prime }}{g}$$

The logarithmic derivative of the quotient of functions is the difference of their logarithmic derivatives. This follows from the quotient rule:

$$\frac{{(f/g)}^{\prime }}{f/g}=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}\frac{g}{f}=\frac{f^{\prime }}{f}-\frac{g^{\prime }}{g}$$

The logarithmic derivative of the $p$-th power of a function is $p$ times the logarithmic derivative of the function. This follows from the power rule:

$$\frac{{(f^p)}^{\prime }}{f^p}=\frac{p{f}^{p-1}{f}^{\prime }}{f^p}=p\frac{f^{\prime }}{f}$$

The logarithmic derivative of the exponential of a function equals the derivative of a function. This follows from the chain rule:

$$\frac{{\left(e^f\right)}^{\prime }}{e^f}=\frac{e^f{f}^{\prime }}{e^f}=f^{\prime }$$

Using these identities, it is rather easy to compute the logarithmic derivatives of expressions which are presented in factored form. For instance, suppose we want to compute the logarithmic derivative of

$$e^{x^2}\frac{{(x-2)}^3(x-3)}{x-1}.$$

Using our identities, we find that its logarithic derivative is

$$2x+\frac{3}{x-2}+\frac{1}{x-3}-\frac{1}{x-1}.$$

Title	logarithmic derivative
Canonical name	LogarithmicDerivative
Date of creation	2013-03-22 16:47:02
Last modified on	2013-03-22 16:47:02
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	11
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 26B05
Classification	msc 46G05
Classification	msc 26A24
Related topic	ZeroesOfDerivativeOfComplexPolynomial